A Unified Precoding Scheme for Generalized Spatial Modulation

Generalized spatial modulation (GSM) activates <tex-math notation="LaTeX">nₜ </tex-math> out of <tex-math notation="LaTeX">Nₜ~(1 ≤ nₜ < Nₜ) </tex-math> available transmit antennas, and information is conveyed through <tex-math notation="LaTeX">nₜ </tex-math> modulated symbols as well as the index of the <tex-math notation="LaTeX">nₜ </tex-math> activated antennas. GSM strikes an attractive tradeoff between spectrum efficiency and energy efficiency. Linear precoding that exploits channel state information at the transmitter enhances the system error performance. For GSM with <tex-math notation="LaTeX">nₜ=1 </tex-math> (the traditional SM), the existing precoding methods suffer from high computational complexity. On the other hand, GSM precoding for <tex-math notation="LaTeX">nₜ ≥ 2 </tex-math> is not thoroughly investigated in the open literature. In this paper, we develop a unified precoding design for GSM systems, which universally works for all <tex-math notation="LaTeX">nₜ </tex-math> values. Based on the maximum minimum Euclidean distance criterion, we find that the precoding design can be formulated as a large-scale nonconvex quadratically constrained quadratic program problem. Then, we transform this challenging problem into a sequence of unconstrained subproblems by leveraging augmented Lagrangian and dual ascent techniques. These subproblems can be solved in an iterative manner efficiently. Numerical results show that the proposed method can substantially improve the system error performance relative to the GSM without precoding and features extremely fast convergence rate with a very low computational complexity.